Wolfram's 2,3 Turing Machine Not Universal

Fishbat writes "In a cutting message to the Foundations of Mathematics mailing list, Stanford's Vaughan Pratt has pointed out an elementary mistake in the recently announced proof that Wolfram's (2,3) machine is universal." Update: 10/30 04:18 GMT by KD : Ed Pegg Jr. from Wolfram Research points to this response to Dr. Pratt's note, which has been submitted to the FoM mailing list but has not yet appeared there due to moderation.

Comments

The Filter

[eldavojohn]

The author of this e-mail, which is incredibly insightful and articulate asked:

How did an argument containing such an elementary fallacy get through the filter? To be fair, I don't consider this that elementary a fallacy but when you break it down to if x + y = infinity then both x & y are infinity it does sound like a rather obvious pitfall. And, in response to his comment on catching it, it had not yet been through "the filter" as the original story stated:

Smith's proof will be published in the journal Complex Systems. Meaning it had not yet been peer reviewed. But I agree with the author however abrasive he came off (although this is far higher than an elementary mistake) when he said:

Had I pushed my luck my second question would have been, who has verified this proof that has taught an automata theory course at a suitably accredited institution? I can forgive an undergrad prof for missing this, his response was probably just to look at it and encourage the kid to submit it. Afterall, what motive would a professor have to read it?

But what should really be noted is what Wolfram himself is quoted as saying from the initial publishing of this proof:

"I had no idea how long it would take for the prize to be won," said Stephen Wolfram. "It could have taken a year, a decade, or a century. I'm thrilled it was so quick. It's an impressive piece of work." Alright, that last sentence there is pretty damning. I have heard time and time again on Slashdot that Wolfram just took other people's work, that he had people working underneath him & that he didn't actually know what he was talking about in his book. This is some corroborating evidence, in my opinion.

I don't know a lot about finite automata but this whole display has shown that Alex Smith is talented but not the winner of the prize, it's best to accept and seek out all criticisms from your community before publishing & Wolfram is not the genius he makes himself out to be. I don't believe I will ever read "A New Kind of Science" as I have many other books in front of that one on my list.

Sounds like just another step in the learning process for Alex--too bad about the cash but he is only 20 and from the looks of it has a bright and promising future. Quite the embarrassment for Wolfram, however.

The real kicker would be if Wolfram had asked his staff to review the proof and they knew it had an elementary mistake and had told him it was golden. Now that would be poetic justice.

Re:The Filter

[Anonymous Coward]

You misread the post. He said that if x + y = infinity and y is finite, then x must be infinity. This is TRUE for numbers. You cannot apply this by analogy to automata and think it is still true. It is not.

Re:The Filter

[jnana]

Indeed. A prior email in that thread -- by the same author, Pratt -- makes it very clear by giving the example of 2 pushdown automata (PDA). A single PDA by itself is not universal, but the system comprised of 2 PDAs is universal, since each stack can represent one side of the Turing machine tape.

As Pratt states, the fallacy is of the following form: a system comprised of 2 PDAs, PDA A and PDA B, is universal. PDA A alone is not universal. Therefore, PDA B must be universal (because the system as a whole is universal). QED.

Of course, in the actual proof, it was not 2 PDAs, but a 2,3 machine and an encoder (i.e.,"PDA A" == "encoder" and "PDA B" == "2,3 machine").

Re:The Filter

[HeadlessNotAHorseman]

Wolfram's 2,3 Turing Machine Not Universal
HOLLYWOOD - In a shock move, MGM has undercut Universal in its bid for the movie rights to Wolfram's 2,3 Turing Machine. Insiders had predicted that Universal would make the deal to build on the phenomenal success of Wolfram's 2,2 Turing Machine, but it has since become apparent that Universal failed to include an option for all sequels in the original contract. The exact figure offered by MGM is unknown, but is believed to be approximately x + y, and we can confirm that y is a finite number. More details will follow.

Bad Headline

[EvanED]

Wolfram's 2,3 Turing Machine Not Universal

That's not, from my reading, what is true. What is true is that the proof is wrong, which means that it may not be universal, but reverts back to the unknown state.

"from the whoa-not-so-fast-there-big-fella dept"

[ZorbaTHut]

Yeah, speaking of that . . .

"Wolfram's 2,3 Turing Machine Not Universal". What? Where'd you get that? This issue doesn't prove anything of the sort - it merely shows that this proof is invalid. It may be universal, it may not, but we still don't know.

So, ironically - whoa, not so fast there big editor.

Bad romantic consequence

[Hao Wu]

I hope that this young man's girlfriend stands by him. So many women would be driven away from an otherwise suitable boyfriend who is skilled at the maths.

Re:Proven?

[EvanED]

Apparently some Slashdot editors have still to defeat the gargantuan misteries of propositional calculus...

Or, you know, English.

Re:duh

[ispeters]

You sound like a troll since you're so belligerent, but, in case anyone else here is legitimately wondering what it means for a Turing machine to be universal, I'll try to answer.

Basically, a Turing Machine is an abstract "computer"--it's a tape (a skinny piece of paper) that has a start but no end (it's infinitely long, but it has a start), and a read/write head that can zip up and down the tape writing, reading, and erasing symbols on the tape. The Church-Turing Thesis postulates that a computable algorithm is any algorithm that can be computed in a finite number of steps by a Turing Machine. There are some things that look like algorithms and seem like they should be computable but are in fact impossible. The classic example is the Halting Problem.

Anyway, a regular Turing Machine only computes one function--it's a single-purpose machine. A Universal Turing Machine is a Turing Machine that can simulate any other Turing Machine by interpreting a codified description of the other machine. Since every computable function is isomorphic to some Turing Machine and every Turing Machine can be simulated by a Universal Turing Machine, every computable function can be computed by a Universal Turing Machine. The computer you're using to read this is an approximation to a Universal Turing Machine (the RAM would have to be infinite in size to be a proper Turing Machine), and the codified descriptions that it interprets are the binary executables that you run on it.

Hope that helps,

Ian

I rate this proof

[MillionthMonkey]

(Score: 2,3 Possibly Informative)

Vaughn Pratt is confused

[evgalois]

No, Vaughn Pratt is confused. There is a post from on the FOM mailing list that explains the confusion. There are subtle issues concerning the nature of computational universality in the presence of infinite initial conditions. Vaughn Pratt is probably remembering work from the 1950s on computational universality, which does not address these issues. There are different definitions that could be given for computational universality with infinite initial conditions. Alex Smith's proof was verified with a particular, natural, definition that was chosen for the prize. So all is well. The 2,3 Turing machine's universality has not been toppled with one email and the personal opinion of Vaughn Pratt. What has happened, though, is that questions that have not been discussed since the 1950s are (this week) back in vogue again.

Re:Vaughn Pratt is confused

[Ed Pegg]

I'm posting from Wolfram Research. Basically, a message from Vaughan Pratt was posted to the correct spot, the FOM list. Dr. Pratt likely didn't expect his message to get a late night SlashDot level exposure. A response to his message has already been sent to the FOM list, but it is a moderated list, and the response is not visible yet. Here is a copy of the FOM posting from Todd Rowland, from the Wolfram prize committee. http://forum.wolframscience.com/showthread.php?s=&threadid=1472 This is how math is done ... trying to poke holes in proofs.